doi:10.1016/S0169-5983(98)00046-X 
Copyright © 1999 The Japan Society of Fluid Mechanics. Published by Elsevier Science B.V.
Numerical simulations of internal solitary waves with vortex cores
A. Aignera, D. Broutmanb and R. Grimshaw
, 
, a
a Department of Mathematics, Monash University, Clayton, VIC 3168, Australia
b School of Mathematics, UNSW, Sydney, NSW 2052, Australia
Received 18 August 1998;
revised 16 November 1998;
accepted 18 December 1998.
Available online 15 October 1999.
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Abstract
This paper deals with the numerical verification of the theory developed by Derzho and Grimshaw (DG) (1997, Phys. Fluids 9(11), 3378–3385) regarding solitary waves in stratified fluids with recirculation regions. The Boussinesq approximation is made and the stratification is chosen such that the Brunt-Väisälä frequency differs only slightly from uniform stratification. To establish the consistency of the numerical scheme the usual KdV and mKdV solutions are tested first and then the solutions obtained by DG are considered. It is found that these waves remain of permanent form and are stationary when viewed at their corresponding phase speed. The recirculation region remains stagnant to first order as predicted by DG.
Fig. 1. Time evolution of density at depth for μ=0.95μmax, σ=0.01, α3=1 and α2=−1.5 (mKdV outer solution).
Fig. 2. Relative phase speed (c−c0)/c0 and eigenvalue λ1 for the KdV outer solution (bold) and KdV (dashed) solution for 1/π
Amax<1/π+μmax, diamonds denote the numerical results.
Fig. 3. Relative phase speed (c−c0)/c0 and eigenvalue λ1 of the DG for the mKdV outer solution (bold) and mKdV (dashed) solution for 1/πAmax<1/π+μmax, diamonds denote the numerical results.
Fig. 4. Plot of the width for 0<μ<μmax for the KdV (bold) and mKdV (dashed) outer solution (α2=1 and α3=1 resp.)
Fig. 5. Time evolution of density at depth for μ=0.95μmax, σ=0.01, α2=1 and α3=0 (KdV outer solution).
Fig. 6. Density plots for normalized times tn=0,69.89,106.67 and 147.13.
Fig. 9. Time evolution of density at depth for μ=0.95μmax, σ=0.01, α2=−1.5 and α3=1 (mKdV outer solution).
Fig. 10. Density plots for normalized times tn=0,70.24,107.21 and 147.88.
Fig. 7. Plot of density (left) and streamfunction (right) for normalized times tn=0,69.89,106.67 and 147.13 inside of the recirculation region, 41×23 grid points resolution for ρ and 61×23 grid points for ψ.
Fig. 11. Plot of density (left) and streamfunction (right) for normalized times tn=0,70.24,107.21 and 147.88 inside of the recirculation region, 41×23 grid points resolution for ρ and 61×23 grid points for ψ.
Fig. 8. Maximum adverse velocity u at the upper boundary.
Fig. 12. Maximum adverse velocity u at the upper boundary.
Abstract
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